Hard Lefschetz Theorem for Nonrational Polytopes

TL;DR

This paper extends the Hard Lefschetz theorem to the intersection cohomology of general, possibly nonrational, polytopes, broadening its applicability beyond rational polytopes associated with toric varieties.

Contribution

It proves the Hard Lefschetz theorem for the intersection cohomology of arbitrary polytopes, including nonrational ones, without relying on an associated toric variety.

Findings

Hard Lefschetz theorem holds for nonrational polytopes' intersection cohomology

Provides a combinatorial construction for intersection cohomology of nonrational polytopes

Extends geometric results to a broader class of polytopes

Abstract

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a general polytope.